Finite Group Databasebeta

← #27287861

Subgroups — #27287861

Order 13568.

398 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 398 conjugacy classes of subgroups — too many to draw. Only its 93 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#20284512C211yesmaximal, characteristic
2#20273489C211yesmaximal, characteristic, complemented
3#20273373C211yesmaximal, characteristic, complemented
4#20273278C211yesmaximal, characteristic, complemented
5#20273036C211yesmaximal, characteristic
6#20272573C211yesmaximal, characteristic, complemented
7C53×#1804C211yesmaximal, characteristic
8#16811180C2211yescharacteristic
9#16810923—22no—
10#16810913—21no—
11#16810903—21no—
12#16810676C2211yescharacteristic
13#16810676C2211yescharacteristic
14#16810663—21no—
15#16810647C2211yescharacteristic
16#16810628—21no—
17#16810375—22no—
18#16810371—21no—
19#16810368—21no—
20#16810359—22no—
21#16810347—22no—
22C2×C4×D4×C53C2211yescharacteristic
23C2×C4⋊C8×C53C2211yescharacteristic
24C2×#102×C53—21no—
25C53×#382C2211yescharacteristic
26C53×#340—21no—
27C2×#11719917—21no—
28#13133217D411yescharacteristic
29#13133217D411yescharacteristic
30#13133209—21no—
31#13133196—21no—
32#13133196—21no—
33C4×Dic106—21no—
34C4×Dic106—21no—
35#13133176—21no—
36#13133150—22no—
37#13133148—22no—
38C4⋊#88322—21no—
39C4⋊#88322—21no—
40C4⋊#88322—21no—
41C4⋊#88322D412yes—
42#13133138—22no—
43#13133138—22no—
44C22⋊#88322—21no—
45C22⋊#88322D412yes—
46C4×#88322—21no—
47C4×#88322—21no—
48C22×D4×C53D411yescharacteristic
49C23×C4×C53—21no—
50C22×C8×C53—21no—
51C4×D4×C53—21no—
52C4×D4×C53—21no—
53C4×D4×C53—21no—
54C4×D4×C53—21no—
55C2×C4⋊C4×C53—21no—
56C2×C4⋊C4×C53D411yescharacteristic
57C2×C22⋊C4×C53—21no—
58C2×C42×C53C2311yescharacteristic
59C4⋊C8×C53—21no—
60C4⋊C8×C53—21no—
61#102×C53—42no—
62#95×C53—21no—
63#95×C53—21no—
64C24×C53—21no—
65C2×D4×C53—21no—
66C2×D4×C53—21no—
67C2×D4×C53—21no—
68C2×D4×C53—21no—
69C2×D4×C53—21no—
70C2×D4×C53Q1612yes—
71C22×C4×C53—21no—
72C22×C4×C53—41no—
73C22×C4×C53—41no—
74C22×C4×C53—41no—
75C22×C4×C53—82no—
76C22×C4×C53C2×D411yescharacteristic
77C22×C4×C53C2×D411yescharacteristic
78C22×C4×C53#4111yescharacteristic
79C2×Dic106—21no—
80C2×Dic106—21no—
81C2×C8×C53—41no—
82C2×C8×C53—41no—
83C2×C8×C53—41no—
84C4⋊C4×C53—21no—
85C4⋊C4×C53—21no—
86C4⋊C4×C53—41no—
87C4⋊C4×C53—41no—
88C22⋊C4×C53—41no—
89C22⋊C4×C53—41no—
90C42×C53C2×D411yescharacteristic
91C42×C53C2×D411yescharacteristic
92C42×C53#4111yescharacteristic
93C42×C53#4111yescharacteristic
94C22⋊Dic53—41no—
95C4⋊Dic53—21no—
96C4⋊Dic53—21no—
97C4⋊Dic53—21no—
98C4⋊Dic53—21no—
99#11719919—41no—
100#11719919—41no—
101C4×Dic53—41no—
102#11719917—41no—
103#11719917—41no—
104C2×#88322—22no—
105C2×#88322—22no—
106C2×#88322—41no—
107C23×C53—22no—
108C23×C53—41no—
109C23×C53—41no—
110C23×C53—81no—
111C23×C53—81no—
112C23×C53#12111yescharacteristic
113D4×C53—22no—
114D4×C53—22no—
115D4×C53—41no—
116D4×C53—41no—
117C2×C4×C53—41no—
118C2×C4×C53—41no—
119C2×C4×C53—41no—
120C2×C4×C53—82no—
121C2×C4×C53—82no—
122C2×C4×C53—81no—
123C2×C4×C53—81no—
124C2×C4×C53—81no—
125C2×C4×C53—81no—
126C2×C4×C53#12011yescharacteristic
127C2×C4×C53#12111yescharacteristic
128C2×C4×C53#12111yescharacteristic
129C2×C4×C53#12311yescharacteristic
130C2×C4×C53#12311yescharacteristic
131C2×C4×C53#12411yescharacteristic
132C2×C4×C53C2×Q1611yescharacteristic
133C2×C4×C53#13511yescharacteristic
134C2×C4×C53AGL(1,ℤ/8)11yescharacteristic
135C2×C4×C53#13711yescharacteristic
136C2×Dic53—41no—
137C2×Dic53—41no—
138Dic106—81no—
139Dic106—81no—
140C8×C53—81no—
141#88322—42no—
142#88322—42no—
143#88322—81no—
144#10469—531nomaximal, Sylow
145C22×C53—21no—
146C22×C53—21no—
147C22×C53—42no—
148C22×C53—82no—
149C22×C53—82no—
150C22×C53—81no—
151C22×C53—81no—
152C22×C53#39411yescharacteristic
153C22×C53#46711yescharacteristic
154C22×C53#46811yescharacteristic
155C4×C53—21no—
156C4×C53—21no—
157C4×C53—21no—
158C4×C53—21no—
159C4×C53—81no—
160C4×C53—161no—
161C4×C53#44911yescharacteristic
162C4×C53#45111yescharacteristic
163C4×C53#48411yescharacteristic
164C4×C53#48511yescharacteristic
165Dic53—81no—
166Dic53—81no—
167#3077—531no—
168#1804D5311yescharacteristic
169#1571—531no—
170#1542—531no—
171#1518—531no—
172#1460—531no—
173#1286—531no—
174C2×C53—21no—
175C2×C53—21no—
176C2×C53—82no—
177C2×C53#203611yescharacteristic
178C2×C53#204811yescharacteristic
179C2×C53#205311yescharacteristic
180#533—531no—
181C2×C4×D4C2×D5311yescharacteristic
182#478—1062no—
183#476—1061no—
184#474—1061no—
185#431—531no—
186#431—531no—
187#428—1061no—
188#423—531no—
189C2×C4⋊C8C2×D5311yescharacteristic
190#418—1061no—
191C2×#102—21no—
192#382C2×D5311yescharacteristic
193#340—21no—
194#335—1062no—
195#333—1061no—
196#332—1061no—
197#328—1062no—
198#323—1062no—
199C53#1046911yescharacteristic, complemented, Sylow
200C22×D4C22⋊D5311yescharacteristic
201C23×C4—21no—
202C2×M16—1061no—
203C22×C8—21no—
204C4⋊Q8—531no—
205C4⋊Q8—531no—
206#125—1061no—
207C22⋊Q8—1061no—
208C22⋊Q8—1061no—
209C4×Q8—1061no—
210C4×Q8—1061no—
211C4×D4—21no—
212C4×D4—21no—
213C4×D4—21no—
214C4×D4—21no—
215#117—1061no—
216C2×C4⋊C4—21no—
217C2×C4⋊C4C22⋊D5311yescharacteristic
218C2×C22⋊C4—21no—
219C2×C42C22×D5311yescharacteristic
220#107—1062no—
221#106—1062no—
222C4⋊C8—21no—
223C4⋊C8—21no—
224C4⋊C8—532no—
225C4⋊C8—1061no—
226C4⋊C8—1061no—
227C4⋊C8—1061no—
228#102—42no—
229#102—1062no—
230#102—1062no—
231C22⋊C8—532no—
232C22⋊C8—1061no—
233C4×C8—1061no—
234C4×C8—1061no—
235#95—21no—
236#95—21no—
237C24—21no—
238C2×Q8—1061no—
239C2×Q8—1061no—
240C2×D4—21no—
241C2×D4—21no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4—21no—
245C2×D4#1171992512yes—
246C22×C4—21no—
247C22×C4—41no—
248C22×C4—41no—
249C22×C4—41no—
250C22×C4—82no—
251C22×C4D4×D5311yescharacteristic
252C22×C4#1171994711yescharacteristic
253C22×C4C2×C22⋊D5311yescharacteristic
254M16—2121no—
255M16—2121no—
256C2×C8—41no—
257C2×C8—41no—
258C2×C8—41no—
259C2×C8—1062no—
260C2×C8—1062no—
261C2×C8—2121no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—41no—
265C4⋊C4—41no—
266C4⋊C4—1061no—
267C4⋊C4—1061no—
268C4⋊C4—1061no—
269C4⋊C4—1061no—
270C4⋊C4—2121no—
271C4⋊C4—2121no—
272C22⋊C4—41no—
273C22⋊C4—41no—
274C22⋊C4—2121no—
275C42—2121no—
276C42#1171994511yescharacteristic
277C42#1171994511yescharacteristic
278C42D4×D5311yescharacteristic
279C42D4×D5311yescharacteristic
280C23—22no—
281C23—41no—
282C23—41no—
283C23—81no—
284C23—81no—
285C23#1313318911yescharacteristic
286Q8—4241no—
287Q8—4241no—
288D4—22no—
289D4—22no—
290D4—41no—
291D4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—81no—
298C2×C4—81no—
299C2×C4—81no—
300C2×C4—81no—
301C2×C4—2121no—
302C2×C4—2121no—
303C2×C4#1313318511yescharacteristic
304C2×C4#1313319011yescharacteristic
305C2×C4#1313319011yescharacteristic
306C2×C4#1313319911yescharacteristic
307C2×C4#1313320011yescharacteristic
308C2×C4#1313320311yescharacteristic
309C2×C4C2×#1171992511yescharacteristic
310C2×C4#1313323711yescharacteristic
311C2×C4#1313324311yescharacteristic
312C2×C4#1313324711yescharacteristic
313C8—81no—
314C8—4241no—
315C8—2122no—
316C8—2122no—
317C22—21no—
318C22—21no—
319C22—42no—
320C22—82no—
321C22—82no—
322C22—81no—
323C22—81no—
324C22#1681053811yescharacteristic
325C22#1681086211yescharacteristic
326C22#1681086711yescharacteristic
327C4—21no—
328C4—21no—
329C4—21no—
330C4—21no—
331C4—81no—
332C4—161no—
333C4—4241no—
334C4—4241no—
335C4#1681076611yescharacteristic
336C4#1681077811yescharacteristic
337C4#1681095811yescharacteristic
338C4#1681096411yescharacteristic
339C2—21no—
340C2—21no—
341C2—82no—
342C2#2027591811yescharacteristic
343C2#2027598611yescharacteristic
344C2#2027601911yescharacteristic
subgroupG/H
centerC22#16810538
commutator, cosocleC2×C42×C53C23
FrattiniC2×C42C22×D53
FittingC53×#1804C2
socleC22×C53#394
prime radicalC22×D4×C53D4
hypercenter#1804D53
hypocenterC53#10469
LCSG ▷ C2×C42×C53 ▷ C22×C53 ▷ C53
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1804

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #20273489C2semidirect
3. #20273373C2semidirect
4. #20273278C2semidirect
6. #20272573C2semidirect
199. C53#10469semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#202845122
#202734892
#202733732
#202732782
#202730362
#202725732
C53×#18042
#1046953